Fun with Volume

We'd been looking at volume and capacity in class for a few days and I came across an activity in an old book that I bought for $2 at the Lifeline bookfair.

 Mathematics for Elementary Teachers - an activity approach

 The Task

So the task was to get a 16 x 16 grid and make a series of regular prisms that have a square base. 


Here's what it looked like:

Shape 1 = 14cm x 14cm x 1cm 

Shape 2 = 12cm x 12 cm x 2cm

Shape 3 = 10cm x 10cm x 3cm etc

The kids were able to see a pattern emerging. Each time you increased the height by 1cm, the sides of the base decreased by 2cm.

They were able to construct a series of 7 shapes:

 Yes - it got a bit messy...

 Eventually we all had a set of shapes.

What Next?

The task then was to arrange these shapes in order from smallest to largest without actually measuring them or doing any calculations.

Here's what a few pictures of what the kids thought...

To help us discuss the activity, we decided to name each shape after its height. So Shape 1 has a height of 1cm, Shape 2 had a height of 2cm etc.

Here are a few comments from the kids about what they noticed when they tried to order the shapes: 
 
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We think they are all the same.

Shape 1 would be the largest because it is long and has the space of the tallest one. I’m just wondering if they would almost be the same since they has the same squares building it.

I reckon if you have a smaller base the sides get bigger. If you get a bigger base you get smaller sides. The one with the biggest base is the biggest shape because you are losing squares each time you fold the edge up.

I’m not exactly sure about Shape 6

Having the biggest base doesn’t mean you have the biggest capacity. Just like if you have the tallest height it isn’t going to have the biggest capacity.

I found this really interesting because I thought that shape 4 was going to be the biggest.  I also found it interesting that we used the same sized piece of paper for every shape.

I found that the largest shape in length isn’t necessarily the shape that has the largest volume.

I found it interesting because the capacity of the shape went up then down. I thought that Shape 4 would be the biggest because it had a big area (base) and was quite high.

I found that the tallest shape will not necessarily have the larger amount of volume.

Doing Some Calculations

So finally we decided to do actually work out the capacity of each shape. Some kids chose to use some of the base-10 blocks to work this out.

 
When we had finished, we drew up a table of data:

Now this was really interesting. Starting with Shape 1, the capacity increases for Shapes 2 and 3, then it starts to go down. This is not what any of us was expecting.

Then we began to wonder.

Is the height of 3cm going to give us the biggest capacity?

What if we try 3.1cm? Or 2.9cm? Will they both be smaller than 3cm?

So we did a bit a table:

And constructed a graph of this data:

And that was really interesting.

And the kids started asking, "What if we try somewhere between 2.6cm and 2.7cm? What about 2.65cm?"

Maybe you can try that one at home....

Don't you love the inquiry mindset?







 







 







 


 

Find the Volume of Irregular Objects

This was my last day at Aoki-Chuo school in Kawaguchi. My visit has been brief but has had a huge impact on my thinking - in fact I'm still thinking about my thinking, if that makes sense.

Anyway, today I was privileged to sit in on several fantastic lessons but I really want to share this one with you.

Topic - Finding Volume of Irregular 3D Shapes

Link to Previous Learning:

The class had been looking previously at how to work out the volume of regular cubes and rectangular prisms. The teacher presented this diagram:

There was a brief discussion about this shape along the lines of how it was similar to and different from the previous shapes that the class had worked with. Everyone seemed happy with what they could see.


Problem Solving Strategies:

Then the teacher asked them for some strategies to deal with this shape.

Two students had ideas:

1. You can separate the shape into smaller parts

2. You can add different bits of the shape together

Even though expressed differently, I think both ideas showed that the kids were looking at the composition of the shape and demonstrated their knowledge of how smaller shapes can be combined to make bigger shapes. This is something that they had done previously in Year 4 with 2-dimensional shapes.


Working Independently:

The class quickly got down to work. To save time, the teacher had prepared multiple copies of the diagram, drawn to a scale of 1:1. This was really helpful - no time spent on doing activities that were not related to the topic. Yes - drawing shapes is a great skill to learn - but not in this lesson. The focus here was always going to be on finding the volume.

The teacher circulated and talked with different children. Once they had a solution, they were encouraged to get another copy of the diagram and work it out another way.


Presentation:

The teacher selected several children who had different strategies and asked them to prepare a presentation for the class. This involved getting some A3 paper with a copy of the diagram (already prepared by the teacher) to write up their solution.



Method 1:

A simple cut divides the shape into two manageable pieces. Works well and gets an accurate solution. 

Note the use of colour in the diagram to highlight the relevant calculations. This was done by the teacher after the presentation when she was clarifying with the students what was going on.


Method 2:

In this solution, the student decided to chop off the top part and add it to the end of the remaining rectangular prism. This works neatly because the base of the chopped off part is 4cm x 6cm, same as the end of the bigger rectangular prism. Neat match! 

There was some confusion and almost disbelief - some students wanted clarification on how this worked.

So the teacher pulled out a model of the shape to show them.

I think she was holding back on showing this model because she didn't want to shape the children's thinking. Personally, I think I might have got the model out earlier or at least got the kids to make the model for themselves BUT making models wasn't the point of the lesson - everything points back to the topic: "Finding the volume of irregular 3D shapes."




Method 3:

This method starts by working out the volume if there was no missing part and subtracting the piece that is gone. Notice once again the use of colour (by the teacher) to show which part of the calculation relates to which part of the diagram.


Same and Different:

This part of the conversation is always interesting. 

The two things that children identified were:

1. You had to use multiplication

2. You had to find rectangular prisms


Generalisation:

"You can find these volumes by looking for cubes and rectangular prisms." 

- a very rough translation from the Japanese

So the students now had a strategy to deal with similar problems. The focus wasn't on getting the number answer correct, in fact it was only peripheral to the conversation.

AND

THE WAS NO MENTION ANYWHERE OF A FORMULA.

I think this was pretty significant - that's why I put it in caps